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<p>In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa–Holm and Novikov equations are ill-posed in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(B_{\infty ,1}^{1}\)</EquationSource> </InlineEquation>. We also construct a new example for initial data and present a short proof of the ill-posedness for the periodic Camassa–Holm equation in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(B^{1+1/p}_{p,r}\)</EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((p,r)\in (1,\infty ]\times (1,\infty ]\)</EquationSource> </InlineEquation>.</p>

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Ill-posedness for the periodic Camassa–Holm type equations in critical Besov spaces

  • 期刊论文

  • 发表日期：

    2025年11月21日

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  • Jinlu Li,
  • Yanghai Yu,
  • Weipeng Zhu

摘要

In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa–Holm and Novikov equations are ill-posed in \(B_{\infty ,1}^{1}\) . We also construct a new example for initial data and present a short proof of the ill-posedness for the periodic Camassa–Holm equation in \(B^{1+1/p}_{p,r}\) with \((p,r)\in (1,\infty ]\times (1,\infty ]\) .